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| The vorticity curl v of a viscous incompressible fluid flow with velocity v plays a prominent role in regularity theory as well as for approximation methods to the Navier-Stokes equations. Of special interest is the creation of vorticity for product formula approaches based on transport-diffusion splitting schemes, which require the alternating change from slip- to no-slip boundary condition.
In order to get bounds to the resulting change of vorticity, recently we had introduced a suitable orthogonal decomposition of the Hilbert space $W^{1,2} (\Omega)$ with respect to the quadratic form $\langle$curl u, curl v$\rangle$ on 3-dimensional domains $\Omega$ with $C^2$-smooth boundaries. The decomposition leads immediately to a lower bound for the considered change of vorticity.
In my talk, this result will be extended to domains having less regular boundaries. In addition, on domains with $C^2$-regular boundaries, we present a transport-diffusion splitting scheme which is consistent with the full Navier-Stokes equations under no-slip condition, and we get also an upper bound to the change of vorticity by transport-diffusion stepping. |
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